To generate points to create the parabola given by the equation \( y = -6x^2 + 3x + 8 \), we can select specific values for \( x \) and calculate the corresponding \( y \) values. Here are a few points calculated for various \( x \) values:
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For \( x = -1 \): \[ y = -6(-1)^2 + 3(-1) + 8 = -6(1) - 3 + 8 = -6 - 3 + 8 = -1 \] Point: \((-1, -1)\)
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For \( x = 0 \): \[ y = -6(0)^2 + 3(0) + 8 = 0 + 0 + 8 = 8 \] Point: \((0, 8)\)
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For \( x = 1 \): \[ y = -6(1)^2 + 3(1) + 8 = -6(1) + 3 + 8 = -6 + 3 + 8 = 5 \] Point: \((1, 5)\)
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For \( x = 2 \): \[ y = -6(2)^2 + 3(2) + 8 = -6(4) + 6 + 8 = -24 + 6 + 8 = -10 \] Point: \((2, -10)\)
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For \( x = 0.5 \): \[ y = -6(0.5)^2 + 3(0.5) + 8 = -6(0.25) + 1.5 + 8 = -1.5 + 1.5 + 8 = 8 \] Point: \((0.5, 8)\)
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For \( x = -2 \): \[ y = -6(-2)^2 + 3(-2) + 8 = -6(4) - 6 + 8 = -24 - 6 + 8 = -22 \] Point: \((-2, -22)\)
So, we have generated the following points on the parabola:
- \((-1, -1)\)
- \((0, 8)\)
- \((1, 5)\)
- \((2, -10)\)
- \((0.5, 8)\)
- \((-2, -22)\)
You can plot these points to visualize the parabola. Additionally, since the leading coefficient is negative, this parabola opens downward.
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